Discussing possible 'abolition movements' is rather difficult in Ancient Greece (particularly Archaic and Classical Greece) as slavery was so embedded in Greek society, and the writers whose opinions we have to reconstruct Ancient Greek attitudes are those who are most likely going to profit greatly from slaves - the elite. However, we do have glimpses of 'abolitionist' sentiment (be aware that abolitionist is an anachronistic term).

In my opinion, the first attestation of such a belief is in Solon. In his poetry he laments how "many of the poor folk find themselves in foreign lands, sold into slavery and bound in shameful bonds" (fr. 4 West), and he then brought back to Attica "many... sold abroad illegally or legally" (fr. 36 West). While this does look like an abolitionist attitude, it is chauvinistic abolitionism, for Solon is only concerned with ensuring that Athenians are no longer enslaved by Athenians, a symptom of the civil strife that he sought to alleviate, rather than concerned with the well-being of all slaves. Solon's concern, first and foremost, was to end the stasis that plagued late-seventh and early-sixth century Athens, it would be foolish to imagine his concern extended to all slaves within Attica.

As far as I am aware, the next serious instance of what could be considered an abolitionist stance is in Euripides' Hecuba, the Chorus Leader remarks "Alas! how cursed is slavery always in its nature, forced by the might of the stronger to endure unseemly treatment" (332-3). This could simply be interpretted as a general expression of sympathy for enslaved women that is evident even in Homer (Od. 8.521-30). However, elsewhere Euripides actually comments on the equality of man, and how the only shame slavery brings is the name itself (Ion, 854-6). How prevalent this view was in fifth ccentury Athens is impossible to say, Euripides might have been considered controversial, particularly when we consider how slaves were depicted in contemporary Athenian comedy, with some scenes depending upon "sadistic amusement at the expense of slaves" (Hunt, 2011: 30). That said, there does appear to be a recognition that those made into slaves are not so because of any particular fault of character, as Plato wrote (Laws, 776d-777a):

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Athenian:
We know, of course, that we would all agree that one ought to own slaves that are as docile and good as possible; for in the past many slaves have proved themselves better in every form of excellence than brothers or sons, and have saved their masters and their goods and their whole houses. Surely we know that this language is used about slaves?

Megillus:
Certainly.

Athenian:
And is not the opposite kind of language also used,—that the soul of a slave has no soundness in it, and that a sensible man should never trust that class at all? And our wisest poet, too, in speaking of Zeus, declared that "Of half their wits far-thundering Zeus bereaves those men on whom the day of bondage falls." Thus each party adopts a different attitude of mind: the one places no trust at all in the servant-class, but, treating them like brute beasts, with goads and whips they make the servants' souls not merely thrice but fifty times enslaved; whereas the other party act in precisely the opposite way.

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Certainly, it appears that some form of philosophical debate about the nature of slavery was going on, and it is likely that the liberation of Messenia and the Helots there only further spurred on this debate. Alkidamas' 'Messenian Oration' argued for the release of the Messenians (Arist. Rhet. 1373b), and Aristotle records the existence of a school of thought that all men were equal and thus slavery was unnatural (Arist. Pol. 1253b.20). However, how prevalent this belief was, and how much people acted upon it, is impossible to surmise from our surviving sources.

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Reference:

P. Hunt, ‘Slaves in Greek Literary Culture’, in K. Bradley, and P. Cartledge (eds.) The Cambridge World History of Slavery, Volume 1: The Ancient World (Cambridge, 2011), 22-47.